Calculus mixed test 3

1. 1
   Rewrite the sign

   Move the negative sign outside the numerator.

   $$-\frac{e^{-3x}-1}{x}$$
2. 2
   Use the derivative limit

   The derivative of e to the negative 3x at zero is negative 3.

   $$-(-3)=3$$

Independent check: L'Hospital's rule gives 3e^(-3x), whose limit at zero is 3.

1. 1
   Apply the quotient rule

   Keep the subtraction order fixed.

   $$f'(x)=\frac{2x(x+2)-(x^2-1)}{(x+2)^2}$$
2. 2
   Simplify

   Combine like terms in the numerator.

   $$f'(x)=\frac{x^2+4x+1}{(x+2)^2}$$

Independent check: Polynomial division gives f(x) = x - 2 + 3/(x + 2), which differentiates to an equivalent expression.

1. 1
   Take logarithms

   Bring the variable exponent down.

   $$\ln y=x\ln x$$
2. 2
   Differentiate implicitly

   Use the product rule on x ln x.

   $$\frac{y'}y=\ln x+1$$
3. 3
   Solve for y prime

   Replace y with x to the x.

   $$y'=x^x(\ln x+1)$$

Independent check: Writing xˣ as e^(x ln x) and using the chain rule gives the same derivative.

1. 1
   Find critical numbers

   Set the derivative equal to zero.

   $$f'(x)=3x^2-3=3(x-1)(x+1)$$
2. 2
   Use the second derivative

   Evaluate 6x at each critical number.

   $$f''(-1)=-6<0,\qquad f''(1)=6>0$$
3. 3
   Classify

   Negative concavity gives a local maximum and positive concavity gives a local minimum.

   $$x=-1:\max,\qquad x=1:\min$$

Independent check: A first-derivative sign chart gives the same increasing and decreasing pattern.

1. 1
   Introduce a limit

   Use a finite upper endpoint first.

   $$\lim_{b\to\infty}\int_1^b x^{-3}dx$$
2. 2
   Evaluate

   The antiderivative is negative one over twice x squared.

   $$\lim_{b\to\infty}\left(\frac12-\frac{1}{2b^2}\right)=\frac12$$

Independent check: With u = 1/x, x⁻³ dx becomes -u du and the bounds reverse, giving ∫₀¹u du = 1/2.

1. 1
   Find an antiderivative

   The derivative of negative cosine is sine.

   $$F(x)=-\cos x$$
2. 2
   Evaluate

   Subtract the lower endpoint value.

   $$-\cos\pi-(-\cos0)=1-(-1)=2$$

Independent check: Composite Simpson's rule with eight equal subintervals gives approximately 2.00027, independently agreeing with the exact value 2.

1. 1
   Differentiate the upper contribution

   Evaluate at 2x and multiply by 2.

   $$2[1+(2x)^3]=2(1+8x^3)$$
2. 2
   Differentiate the lower contribution

   Evaluate at x squared, multiply by 2x, and subtract.

   $$2x[1+(x^2)^3]=2x(1+x^6)$$
3. 3
   Combine

   Preserve the lower-bound minus sign.

   $$A'(x)=2(1+8x^3)-2x(1+x^6)$$

Independent check: Evaluating the polynomial antiderivative at both bounds and differentiating gives the same expression.

1. 1
   List the derivative values

   All derivatives of e to the x equal e to the x.

   $$f^{(n)}(0)=1$$
2. 2
   Build the polynomial

   Divide each coefficient by its factorial.

   $$P_3(x)=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}$$

Independent check: P3 and eˣ have the same value and first three derivatives at x = 0.

1. 1
   Separate and integrate

   Move y to the left and x terms to the right.

   $$\int\frac{dy}{y}=\int2x\,dx$$
2. 2
   Exponentiate

   Convert the logarithmic equation to an exponential family.

   $$y=Ce^{x^2}$$
3. 3
   Apply the initial value

   At x = 0, e to the zero is 1.

   $$3=C$$

Independent check: Differentiating gives y′ = 6xe^(x²) = 2xy, and y(0) = 3.

1. 1
   Write the disk area

   The radius is y = square root x.

   $$A(x)=\pi(\sqrt{x})^2=\pi x$$
2. 2
   Integrate

   Accumulate the disk areas from 0 to 4.

   $$V=\pi\left[\frac{x^2}{2}\right]_0^4=8\pi$$

Independent check: Horizontal shells about the x-axis give ∫₀²2πy(4 - y²) dy = 8π, confirming the disk-method result by a second setup.